The partition function is Z = Σ c o n f e − β H = Σ S p c P ( S p c ) , where β = 1 / ( k B T ) . Following the standard ERR procedure [27], the unnormalize probability for a central spin s o on sublattice A is

where Q n ( s 0 | σ 1 ( k ) ) is the partition function of the k-th branch of generation nstarting from the nearest-neighbor spin σ 1 ( k ) on sublattice B, computed recursively as

with p = q − 1 branches per neighbor spin. Defining the branch partition function on sublattice A as f n ( { s 0 } ) = ∑ { σ 1 } Q n ( s 0 | σ 1 ) , Equation (2.3) yields

Considering all ten spin values of sublattice B ( σ 1 ∈ { ± 9 / 2 , ± 7 / 2 , ± 5 / 2 , ± 3 / 2 , ± 1 / 2 } ), Equation (2.4) expands to

Substituting s 0 = + 3 / 2 , − 3 / 2 , + 1 / 2 , − 1 / 2 into Equation (2.5) yields directly the four branch partition functions f n ( ± 3 / 2 ) and f n ( ± 1 / 2 ) used in the numerical fixed-point iteration. Their explicit forms follow immediately and are not reproduced separately, since each is a special case of Equation (2.5).

Similarly, the branch partition function for sublattice B is

Evaluating Eq. (eq:fn_sigmaB 2.6) for each of the ten spin states σ 1 ∈ { ± 9 / 2 , ± 7 / 2 , ± 5 / 2 , ± 3 / 2 , ± 1 / 2 } provides the ten branch partition functions f n − 1 ( σ 1 ) entering Equation (2.5). Again, each is a direct substitution into Equation (2.6) and the five resulting expressions are used directly in the iteration without being reproduced in full.

For sublattice A (spin-3/2), three independent ratios are defined with respect to the reference state s 0 = − 1 / 2 :

For sublattice B (spin-9/2), ten branch partition functions arise. Nine independent ratios are defined with respect to the reference state σ 1 = − 1 / 2 (note Y 6 ≡ 1 by construction):

Numerical implementation and reproducibility. All fixed-point equations are iterated in the log-ratio representation ℓ X k = ln X k ( k = 1 , 2 , 3 ) and ℓ Y l = ln Y l ( l = 1 , ⋯ , 5 ,   7 , ⋯ , 10 , with Y 6 ≡ 1 excluded), which prevents arithmetic overflow at low temperatures. All log-sum-exp operations are stabilised as ln ∑ i e a i = max i a i + ln ∑ i e a i − max i a i .

Ground-stateinitialisation. For each parameter set ( D A , D B , q , J ) , initial log-ratios are provided by a dedicated ground-state routine that runs the ERR at T gs = 0.05   | J | / k B from all 4 × 10 = 40 pure-spin initialisations (every combination of the four s A states and ten σ B states), using a strict convergence tolerance ε gs = 10 − 14 and a maximum of N gs = 20   000 iterations. The internal energy is computed at each resulting fixed point, and the log-ratios of the minimum-energy fixed point are taken as the starting values for the subsequent temperature (or field) sweep. Among fixed points of comparable energy, preference is given to those with M A > 0 to ensure consistency of sign conventions across the phase diagram.

Main temperature and field sweeps. During the sweep, the converged solution at step n is used directly as the initial guess for step n + 1 (continuation method), so that the iteration tracks a given solution branch continuously rather than always relaxing to the global free-energy minimum. This is essential for following metastable branches (first-order transitions, hysteresis). The iteration is stopped when max k | Δ ℓ X k | < ε = 10 − 10 , with a hard limit of N max = 5000 iterations per step.

The sublattice magnetization M A is the thermal average of s 0 :

Similarly, M B is

where Z B is the corresponding normalization sum (denominator with all positive signs and the same exponential pre-factors).

Second-order transitions occur when both sublattice magnetizations vanish continuously. At the critical temperature T c , the conditions are

where β c = 1 / ( k B T c ) . These conditions are satisfied in the paramagnetic phase through the spin-reversal symmetry relations

Substituting these conditions into the fixed-point equations yields the self-consistent relations at T c :

where ( α 1 , ⋯ , α 5 ) = ( 81 / 4 ,   49 / 4 ,   25 / 4 ,   9 / 4 ,   1 / 4 ) and ( γ 1 , ⋯ , γ 5 ) = ( 27 / 4 ,   24 / 4 ,   15 / 4 ,   9 / 4 ,   3 / 4 ) , with Y 6 * = 1 . The symmetric Y-ratios satisfy

with C 1 = cosh ( 3 / 2 δ k β c J ) , C 2 = cosh ( 1 / 2 δ k β c J ) , for k = 1 , 2 , 3 , 4 , where ( δ 1 , δ 2 , δ 3 , δ 4 ) = ( 9 / 2 , 7 / 2 , 5 / 2 , 3 / 2 ) are the spin values of sublattice B. These self-consistent equations are solved numerically for each parameter set ( D A , D B , q ) to construct the phase diagrams of Section 3.

The total magnetization per site is defined as

The compensation temperature T comp , when it exists, satisfies

To map out the stability regions, we calculate the phase diagram in the ( D A / q | J | , D B / q | J | ) plane at h = 0 . The ground states minimize the energy per site

Restricting to ferrimagnetic configurations, we compare all ten relevant spin pairs: (3/2, −9/2), (3/2, −7/2), (3/2, −5/2), (3/2, −3/2), (3/2, −1/2), (1/2, −9/2), (1/2, −7/2), (1/2, −5/2), (1/2, −3/2), (1/2, −1/2). A complete enumeration of all 4 × 10 = 40 spin-pair combinations confirms that, for J < 0 , the minimum-energy configurations are always of the ferrimagnetic type (opposite-sign spins on the two sublattices); ferromagnetic pairs (same-sign spins) and configurations with one zero-spin component have strictly higher ground-state energies throughout the entire ( D A / q | J | , D B / q | J | ) plane explored here, so the restriction to the ten antiparallel pairs listed above is exact.

The resulting ground-state diagram is shown in Figure 1. Seven multicritical points (A-G) emerge where three or more phases become degenerate; their coordinates are listed in Table 2.

The diagram reveals extended stability domains for high-spin states on sublattice B, particularly (3/2; −9/2) and (3/2; −7/2), when D B / q | J | takes strongly negative values. Compared with the (3/2, 7/2) case [27] (five multicritical points), the (3/2, 9/2) model has seven such points, reflecting the additional competing ground-state configurations introduced by the spin-9/2 sublattice. This result is in excellent agreement with the ground-state diagram reported in [28] for the same spin pair on a simple cubic lattice ( z = 6 ).

The magnetization curves were obtained by numerically solving the ERR on the Bethe lattice with q = 4 and h / | J | = 0 .

Figure 1. Ground-state phase diagram of the model in the ( D A / q | J | , D B / q | J | ) plane ( q is the coordination number).

Table 2.Coordinate of the multicritical points in the ground-state phase diagram.

Identification of transition types and compensation points.Second-order(continuous) transitions are identified as the temperature at which the order parameter | M A | + | M B | vanishes continuously. Numerically, T c is located at the last temperature step where | M A | + | M B | > 0.02 , with linear interpolation to the threshold; the threshold value 0.02 is chosen to be well above numerical noise while small enough not to bias the estimate. First-order(discontinuous) transitions are identified by a magnetization jump exceeding Δ M > 0.08 between two successive temperature steps along the continuous solution branch; they are further confirmed by the simultaneous existence of two distinct stable fixed points (the ground-state branch and a competing metastable branch) initialized independently for the same parameters. The precise location of first-order boundaries carries the numerical uncertainty of the temperature step used ( δ T ≈ 0.01 | J | / k B in the relevant panels); no additional free-energy comparison between branches is performed. Compensation temperatures are detected as sign changes of M T = 1 / 2 ( M A + M B ) occurring while | M A | > 0.02 and | M B | > 0.02 simultaneously, confirmed by linear interpolation between the bracketing temperature steps.

Results are shown in Figures 2-5.

Figure 2. Thermal variations of M A and M B for D A / | J | = 4.0 and selected values of D B / | J | .

Figure 2 shows results for D A / | J | = 4.0 . The sublattice A magnetization starts from its saturation value of 3/2 for all D B / | J | , since a large positive crystal field strongly favors s = 3 / 2 . Meanwhile, M B exhibits at T = 0 nine distinct saturation values (−1/2, −1, −3/2, −2, −5/2, −3, −7/2, −4, −9/2) for D B / | J | = − 0.50 through −4.00, respectively—a direct signature of the ten spin states of sublattice B. Both magnetizations vanish continuously at a common T c , indicating second-order transitions throughout.

Figure 3 displays results for D A / | J | = − 8.0 , organized in three panels. Panel (a): for D B / | J | < − 0.36 , M A starts from 1/2 (large negative D A favors s = 1 / 2 ) and second-order transitions occur throughout. Panel (b): for − 0.36 ≤ D B / | J | < − 0.33 , the system exhibits first-order transitions, with a discontinuous jump between two competing ordered phases. Panel (c): for D B / | J | ≥ − 0.33 , a single second-order transition is recovered.

Figure 4 shows results for D B / | J | = 0.4 and three values of D A / | J | . The starting values M A = 1 / 2 , 1 , 3 / 2 for D A / | J | = − 12 , − 9 , 4 are fully consistent with the ground-state diagram. The occurrence of both first- and second-order

Figure 3. Thermal behavior of M A and M B for D A / | J | = − 8.0 and selected values of D B / | J | (three panels).

Figure 4. Thermal variations of M A and M B for D B / | J | = 0.4 and different values of D A / | J | .

transitions is qualitatively consistent with the mean-field results of [28].

To clarify the compensation behavior, we examine M T = 1 / 2 ( M A + M B ) for D B / | J | = − 2.0 (Figure 5). All curves except D A / | J | = − 0.8 pass through a compensation point before vanishing at T c . The compensation behavior is only weakly affected by D A / | J | , while T c increases with D A / | J | . This is qualitatively consistent with [29].

Figure 5. Thermal behaviors of the total magnetization M T for D B / | J | = − 2.0 , q = 4 , and selected D A / | J | . Compensation temperatures are visible for most parameter values.

Figure 6 displays second-order transition lines in the ( D / | J | , k T c / | J | ) plane for the isotropic case D A = D B = D and q = 3 , 4 , 5 , 6 . The critical temperature T c increases as | D | decreases (weaker crystal field favors ordering) and increases with q (more nearest neighbors enhance ordering).

Figure 7 shows the ( D A / | J | , k T c / | J | ) phase diagram. When D B / | J | < − 0.25 (panel a), critical curves behave analogously to Figure 6. For D B / | J | ≥ − 0.25 (panel b), the critical lines originate at high temperatures and grow rapidly as D A / | J | → ∞ . The final phase diagram, in the ( D B / | J | , k T c / | J | ) plane, is presented in Figure 8.

Hysteresis computation protocol. Hysteresis loops are computed by sweeping the external field from h / | J | = − 15 to +15 (forward branch) and back to −15 (return branch), using N h = 600 equally spaced field steps in each direction

Figure 6. Finite-temperature phase diagrams in the ( D / | J | , k T c / | J | ) plane for q = 3 , 4 , 5 , 6 : second-order transition lines.

Figure 7. Phase diagrams in the ( D A / | J | , k T c / | J | ) plane for q = 4 , with selected D B / | J | values.

Figure 8. Phase diagrams in the ( D B / | J | , k T c / | J | ) plane for q = 4 , with selected D A / | J | values, second-order transitions.

( δ h / | J | ≈ 0.05 , i.e. step size 30/599). The forward sweep begins from the fixed point converged at h / | J | = − 15 ; at each subsequent field value, the ERR iteration is initialized from the converged solution of the immediately preceding field step (sequential continuation). This procedure is essential for capturing hysteresis: in bistable regions, the forward and return branches converge to different fixed points, reproducing the physical irreversibility of the magnetization process. The same iteration parameters as the thermal sweeps are used ( N max = 5000 , ε = 10 − 10 ) at every field step.

We examine the effect of temperature, uniform crystal field, and ferrimagnetic coupling J on the hysteresis behavior.

For D A / | J | = 0 , D B / | J | = − 1 (Figure 9), a single hysteresis loop is present; it shrinks and disappears for k T / | J | > 6 , reflecting the progressive loss of magnetic ordering.

Figure 10 shows the hysteresis loops for DA/J = 1 and DB/J = 0 at selected temperatures. The loop area decreases progressively with increasing temperature and vanishes near the critical temperature, indicating the transition from the ferrimagnetic phase to the paramagnetic phase.

Figure 11 shows the effect of the uniform crystal field at k T / | J | = 0.5 . No hysteresis occurs for D / | J | < − 1.9 ; a single loop appears for − 1.9 ≤ D / | J | < − 1.25 and for D / | J | ≥ − 1.25 . The loop width remains nearly constant for D / | J | ≥ 0 , so that neither the coercive field nor the remanent magnetization is significantly affected by further increases in the positive crystal field.

Figure 9. Hysteresis loops for D A / | J | = 0 and D B / | J | = − 1 at selected temperatures.

Figure 10.Hysteresis loops for D A / | J | = 1 and D B / | J | = 0 at selected temperatures.

Figure 11.Influence of the uniform crystal field on the hysteresis behavior at k T / | J | = 0.5 .

Figure 12 shows the effect of the ferrimagnetic coupling J at k T / | J | = 0.5 , D A / | J | = 0 , D B / | J | = − 1 . For J < − 0.6 , only a single loop is observed. In the range − 0.6 ≤ J < − 0.5 , the system exhibits three distinct hysteresis loops—a qualitatively richer behavior than in the (3/2, 7/2) case [27]—before returning to a single loop for J ≥ − 0.45 , which eventually vanishes at J = − 0.25 .

The broadening of the triple-loop regime in the (3/2, 9/2) system compared with (3/2, 7/2) has a clear physical origin rooted in the exchange-crystal-field competition on sublattice B. With spin-9/2, there are nine field-induced level crossings between adjacent σ states (versus seven for spin-7/2), each corresponding to a discontinuous plateau transition in M T ( h ) . At low temperature and intermediate | J | , the antiferromagnetic exchange energy and the crystal-field energy D B compete to stabilise adjacent spin states, so that multiple metastable plateaux remain accessible over a wider range of the ratio J / D B . This competition generates three simultaneously accessible plateaux—hence the triple-loop structure—over the window − 0.6 ≤ J / | J | < − 0.5 , a range that is wider than the analogous window in the spin-7/2 case precisely because the higher-spin sublattice supports a larger number of competing spin-state pairs. The higher spin values σ = ± 9 / 2 also carry a larger magnetic moment than σ = ± 7 / 2 , which strengthens the effective exchange field experienced by sublattice A, raising T c systematically for equivalent reduced parameters. In contrast, for large | J | , the exchange energy dominates and a single strongly antiparallel ground state is stabilised, suppressing the intermediate plateaux and collapsing the multi-loop structure back to a single loop.

Figure 13 quantifies the coercive field h c / | J | and remanent magnetization

Figure 12. Hysteresis loops for D A / | J | = 0 , D B / | J | = − 1 , k T / | J | = 0.5 , and various values of J.

Figure 13. Coercive field h c / | J | and remanent magnetization M R versus J / | J | at D A / | J | = 0 , D B / | J | = − 1 , k T / | J | = 0.5 .

M R as functions of J / | J | . The coercive field decreases almost linearly to zero around J ≈ − 0.25 . The remanent magnetization remains nearly constant (≈0.77) for strongly negative J , then drops sharply near J = − 0.5 , marking the transition from the multi-loop to the single-loop hysteresis regime. Strongly negative J values correspond to hard magnetic behavior (high coercivity and remanence), suitable for permanent magnets and data storage, while values near zero correspond to soft magnetic behavior (low coercivity), desirable for transformers and electromagnetic cores.